Theorem spw 2041

Description: Weak version of the specialization scheme sp 2180. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2180 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2180 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2136 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2180 are spfw 2040 (minimal distinct variable requirements), spnfw 1984 (when 𝑥 is not free in ¬ 𝜑), spvw 1985 (when 𝑥 does not appear in 𝜑), sptruw 1808 (when 𝜑 is true), spfalw 2004 (when 𝜑 is false), and spvv 2003 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1911	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1911	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1911	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2040	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782

This theorem is referenced by:  hba1w  2054  spaev  2057  ax12w  2134  nfcriOLD  2946  reldisj  4359  bj-ssblem1  34100  bj-ax12w  34123